# For what values of x is #f(x)= x^4-3x^3-4x-7 # concave or convex?

the function is convex in

You would analyze the second derivative; the first one is:

then the second one is:

that's

Then the given function is convex in

and concave in

graph{x^4-3x^3-4x-7 [-5, 5, -27, 10]}

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To determine where the function ( f(x) = x^4 - 3x^3 - 4x - 7 ) is concave or convex, you need to find the second derivative of the function and then analyze its sign.

First, find the second derivative: [ f''(x) = 12x^2 - 18x ]

Next, set ( f''(x) ) equal to zero and solve for ( x ): [ 12x^2 - 18x = 0 ] [ 6x(2x - 3) = 0 ] [ x = 0 \quad \text{or} \quad x = \frac{3}{2} ]

Now, you can test intervals created by these critical points with the second derivative test:

- Choose a value of ( x ) less than ( 0 ) (e.g., ( x = -1 )) and plug it into ( f''(x) ) to determine its sign.
- Choose a value of ( x ) between ( 0 ) and ( \frac{3}{2} ) (e.g., ( x = 1 )) and plug it into ( f''(x) ).
- Choose a value of ( x ) greater than ( \frac{3}{2} ) (e.g., ( x = 2 )) and plug it into ( f''(x) ).

Analyzing the signs of ( f''(x) ) in these intervals will determine where the function is concave up (convex) or concave down.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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